Exact SDP relaxations for a class of quadratic programs with finite and infinite quadratic constraints

Abstract

We investigate exact semidefinite programming (SDP) relaxations for the problem of minimizing a nonconvex quadratic objective function over a feasible region defined by both finitely and infinitely many nonconvex quadratic inequality constraints (semi-infinite QCQPs). Specifically, we present two sufficient conditions on the feasible region under which the QCQP, with any quadratic objective function over the feasible region, is equivalent to its SDP relaxation. The first condition is an extension of a result recently proposed by the authors (arXiv:2308.05922, to appear in SIAM J. Optim.) from finitely constrained quadratic programs to semi-infinite QCQPs. The newly introduced second condition offers a clear geometric characterization of the feasible region for a broad class of QCQPs that are equivalent to their SDP relaxations. Several illustrative examples, including quadratic programs with ball-, parabola-, and hyperbola-based constraints, are also provided.